Convolution kernel determination problem for the time-fractional diffusion equation

Abstract

In this work, we study the inverse problem of determining the convolution kernel in the Cauchy problem for the time-fractional diffusion equation by a single observation at the point x=0 of the diffusion process. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation is constructed. In particular, using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel function which will be used for the study inverse problem. The inverse problem is reduced to the equivalent integral equation of the Volterra type. By the contracted mapping principle, the local existence and global uniqueness results are proven. Also the stability estimate is obtained.